Theorem xpdom2g 7085

Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
xpdom2g	|- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )

Proof of Theorem xpdom2g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 4765	. . . . 5 |- ( x = C -> ( x X. A ) = ( C X. A ) )
2		xpeq1 4765	. . . . 5 |- ( x = C -> ( x X. B ) = ( C X. B ) )
3	1, 2	breq12d 4124	. . . 4 |- ( x = C -> ( ( x X. A ) ~<_ ( x X. B ) <-> ( C X. A ) ~<_ ( C X. B ) ) )
4	3	imbi2d 230	. . 3 |- ( x = C -> ( ( A ~<_ B -> ( x X. A ) ~<_ ( x X. B ) ) <-> ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) ) )
5		vex 2818	. . . 4 |- x e. _V
6	5	xpdom2 7084	. . 3 |- ( A ~<_ B -> ( x X. A ) ~<_ ( x X. B ) )
7	4, 6	vtoclg 2877	. 2 |- ( C e. V -> ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) )
8	7	imp 124	1 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   class class class wbr 4111   X. cxp 4749   ~<_ cdom 6976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fv 5362  df-dom 6979

This theorem is referenced by:  xpdom1g  7086